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Properties

Label 2-731-43.35-c1-0-22
Degree $2$
Conductor $731$
Sign $-0.804 - 0.594i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.100 + 0.441i)2-s + (0.452 + 1.98i)3-s + (1.61 + 0.778i)4-s + (−1.88 + 2.36i)5-s − 0.921·6-s + 4.55·7-s + (−1.07 + 1.34i)8-s + (−1.02 + 0.492i)9-s + (−0.853 − 1.07i)10-s + (−2.36 + 1.14i)11-s + (−0.812 + 3.55i)12-s + (0.657 − 0.824i)13-s + (−0.459 + 2.01i)14-s + (−5.53 − 2.66i)15-s + (1.75 + 2.19i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.0713 + 0.312i)2-s + (0.261 + 1.14i)3-s + (0.808 + 0.389i)4-s + (−0.842 + 1.05i)5-s − 0.376·6-s + 1.72·7-s + (−0.379 + 0.475i)8-s + (−0.341 + 0.164i)9-s + (−0.269 − 0.338i)10-s + (−0.714 + 0.343i)11-s + (−0.234 + 1.02i)12-s + (0.182 − 0.228i)13-s + (−0.122 + 0.538i)14-s + (−1.42 − 0.688i)15-s + (0.437 + 0.549i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.804 - 0.594i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.804 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617565 + 1.87510i\)
\(L(\frac12)\) \(\approx\) \(0.617565 + 1.87510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (5.40 + 3.71i)T \)
good2 \( 1 + (0.100 - 0.441i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-0.452 - 1.98i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (1.88 - 2.36i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 - 4.55T + 7T^{2} \)
11 \( 1 + (2.36 - 1.14i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.657 + 0.824i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (1.21 + 0.583i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-5.69 + 2.74i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-2.15 + 9.42i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-2.09 + 9.16i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + 8.68T + 37T^{2} \)
41 \( 1 + (0.894 - 3.91i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (5.11 + 2.46i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-3.80 - 4.76i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (1.79 + 2.24i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (1.34 + 5.87i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-8.09 - 3.89i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (14.3 + 6.88i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-4.56 + 5.72i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 8.31T + 79T^{2} \)
83 \( 1 + (1.43 + 6.28i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-1.18 - 5.19i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-11.5 + 5.57i)T + (60.4 - 75.8i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.65070093053044921292381595897, −10.27407196899803051695826178235, −8.778544905295530119434692071510, −7.968075275751709556021555975691, −7.53911662236588901344469367916, −6.49467963594049045784269609336, −5.14161784472971190384977609366, −4.29003334733376298565428717687, −3.28007280328349293412447530118, −2.24174012060473807596987133622, 1.15007199353965162947573840381, 1.68195415387264706230722779492, 3.12886272824960454402950252600, 4.78037282642256118210070679511, 5.35897083561470122848463774395, 6.92829529260387436799357039171, 7.41184250288150981677411353632, 8.468358539908643265380595704953, 8.610637963676849580940282299949, 10.36492666847300636744385370345

Graph of the $Z$-function along the critical line